Appendix D — Partial derivatives of \(F_i\) to pressure heads

The coefficients of the Jacobian are given by:

\[ \begin{array}{ll} \frac{\partial F_i}{\partial h^{j+1,p}_{i-1}} & = - \frac{K^{j+\kappa,\kappa p}_{i-1/2}} {0.5\left(\Delta z_{i-1}+\Delta z_i\right)} - \kappa \frac{\partial K^{j+\kappa,\kappa p}_{i-1/2}} {\partial h^{j+1,p}_{i-1}} \left(\frac{h^{j+1,p}_{i-1}-h^{j+1,p}_{i}} {0.5\left(\Delta z_{i-1}+\Delta z_i\right)}+1\right)\\ \frac{\partial F_i}{\partial h^{j+1,p}_{i}} & = \frac{\Delta z_i} {\Delta t^j} C_i^{j+1,p} + \Delta z_i \frac{\partial S^{j+1,p}_{m,i}} {\partial h_i^{j+1,p}} + \frac{K^{j+\kappa,\kappa p}_{i-1/2}} {0.5 \left(\Delta z_{i-1} + \Delta z_i\right)} + \frac{K^{j+\kappa,\kappa p}_{i+1/2}} {0.5 \left(\Delta z_{i} + \Delta z_{i+i}\right)}\\ & -\kappa \frac{\partial K^{j+\kappa,\kappa p}_{i-1/2}} {\partial h^{j+1,p}_{i}} \left(\frac{h^{j+1,p}_{i-1} - h^{j+1,p}_{i}}{0.5 \left(\Delta z_{i-1} + \Delta z_{i}\right)}+1\right) +\kappa \frac{\partial K^{j+\kappa,\kappa p}_{i+1/2}} {\partial h^{j+1,p}_{i}} \left(\frac{h^{j+1,p}_{i} - h^{j+1,p}_{i+1}}{0.5 \left(\Delta z_{i} + \Delta z_{i+1}\right)}+1\right)\\ \frac{\partial F_i}{\partial h^{j+1,p}_{i+1}} & = - \frac{K^{j+\kappa,\kappa p}_{i+1/2}} {0.5 \left(\Delta z_{i} + \Delta z_{i+1}\right)} + \kappa \frac{\partial K^{j+\kappa,\kappa p}_{i+1/2}} {\partial h^{j+1,p}_{i+1}} \left(\frac{h^{j+1,p}_{i}-h^{j+1,p}_{i+1}} {0.5 \left(\Delta z_{i} + \Delta z_{i+1}\right)}+1\right) \end{array} \tag{D.1}\]

Where \(C_i^{j+1,p}\)is the differential moisture capacity (cm^-1). \(\frac{\partial K^{j+\kappa,\kappa p}_{i-1/2}}{\partial h^{j+1,p}_{i}}\) and \(\frac{\partial K^{j+\kappa,\kappa p}_{i+1/2}}{\partial h^{j+1,p}_{i}}\) are the partial derivatives of the internodal conductivity to the pressure head. The calculation of the partial derivatives for the top and bottom compartment requires special attention.

The Jacobian coefficient for the first compartment reads as:

Flux controlled top boundary condition

\[ \frac{\partial F_i}{\partial h^{j+1}_{1}} = \frac{\Delta z_1} {\Delta t^j} C_1^{j+1} + \Delta z_1 \frac{\partial S^{j+1}_{m,1}} {\partial h_1^{j+1}} + \frac{K^{j+\kappa}_{1+0.5}} {0.5 \left(\Delta z_{i} + \Delta z_{i+1}\right)} + \frac{\partial K^{j+\kappa}_{1+0.5}} {\partial h^{j+1}_{1}} \left(\frac{h^{j+1}_{1}-h^{j+1}_{2}}{0.5 \left(\Delta z_{1} + \Delta z_{2}\right)}+1\right) \tag{D.2}\]

Head controlled top boundary condition

\[ \begin{array}{ll} \frac{\partial F_i}{\partial h^{j+1}_{1}} = & \frac{\Delta z_1} {\Delta t^j} C_1^{j+1} + \Delta z_1 \frac{\partial S^{j+1}_{m,1}} {\partial h_1^{j+1}} + \frac{K^{j+1}_{1/2}}{0.5\Delta z_i} + \frac{K^{j+\kappa}_{1+0.5}} {0.5 \left(\Delta z_{i} + \Delta z_{i+1}\right)}\\ & -\frac{\partial K^{j+\kappa}_{1/2}} {\partial h^{j+1}_{1}} \left(\frac{h^{j+1}_{0}-h^{j+1}_{1}}{0.5\Delta z_1}+1\right) + \kappa \frac{\partial K^{j+\kappa}_{1+0.5}} {\partial h^{j+1}_{1}} \left(\frac{h^{j+1}_{1}-h^{j+1}_{2}}{0.5\left(\Delta z_1+\Delta z_2\right)}+1\right)\\ \end{array} \tag{D.3}\]

The internodal conductivity \(K_{1/2}^{j+1}\) is irrespective the value of \(\kappa\) always treated implicitly.

The Jacobian coefficient for the last compartment reads as:

Flux controlled bottom boundary

\[ \begin{array}{ll} \frac{\partial F_1}{\partial h^{j+1}_{n}} & = \frac{\Delta z_n} {\Delta t^j} C_n^{j+1} + \Delta z_n \frac{\partial S^{j+1}_{m,n}} {\partial h_n^{j+1}} + \frac{K^{j+\kappa}_{n-1/2}} {0.5 \left(\Delta z_{n-1} + \Delta z_n\right)}\\ &-\kappa \frac{\partial K^{j+\kappa}_{n-1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_{n-1}-h^{j+1}_{n}}{0.5\left(\Delta z_{n-1}+\Delta z_n\right)}+1\right)\\ \end{array} \tag{D.4}\]

Head controlled bottom boundary

\[ \begin{array}{ll} \frac{\partial F_n}{\partial h^{j+1,p}_{n}} & = \frac{\Delta z_n} {\Delta t^j} C_n^{j+1} + \Delta z_n \frac{\partial S^{j+1}_{m,n}} {\partial h_n^{j+1}} + \frac{K^{j+\kappa}_{n-1/2}} {0.5\left(\Delta z_{n-1} + \Delta z_n\right)} + \frac{K^{j+\kappa}_{n+1/2}} {0.5\Delta z_{n}}\\ & -\kappa \frac{\partial K^{j+\kappa}_{n-1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_{n-1} - h^{j+1}_{n}}{0.5\left(\Delta z_{n-1} + \Delta z_n\right)}+1\right) +\kappa \frac{\partial K^{j+\kappa}_{n+1/2}} {\partial h^{j+1}_n} \left(\frac{h^{j+1}_n - h_{bot}}{0.5\Delta z_n}+1\right)\\ \end{array} \tag{D.5}\]

Predefined groundwater levels

\[ \begin{array}{ll} \frac{\partial F_i}{\partial h^{j+1}_{n^*}} & = \frac{\Delta z_i} {\Delta t^j} C_{n^*}^{j+1} + \Delta z_{n^*} \frac{\partial S^{j+1}_{m,n^*}} {\partial h_{n^*}^{j+1}} + \frac{K^{j+\kappa}_{{n^*}-1/2}} {0.5\left(\Delta z_{{n^*}-1} + \Delta z_{n^*}\right)}\\ & -\kappa \frac{\partial K^{j+\kappa}_{{n^*}-1/2}} {\partial h^{j+1}_{n^*}} \left(\frac{h^{j+1}_{n^*-1} - h^{j+1}_{n^*}}{0.5\left(\Delta z_{n^*-1} + \Delta z_{n^*}\right)}+1\right) + \left(\frac{z_{n^*}-z_{n^*+1}}{z_{n^*}-gwl}\right) \frac{K^{j+\kappa}_{{n^*}+1/2}} {0.5\left(\Delta z_{{n^*}} + \Delta z_{n^*+1}\right)}\\ & + \kappa \frac{\partial K^{j+\kappa}_{n^*+1/2}} {\partial h^{j+1}_{n^*}} \left(\left(\frac{z_{n^*}-z_{n^*+1}}{z_{n^*}-gwl}\right)\frac{h^{j+1}_{n^*}}{0.5\left(\Delta z_{n^*}+\Delta z_{n^*+1}\right)}+1\right)\\ \end{array} \tag{D.6}\]

Cauchy relation for the bottom boundary

\[ \begin{array}{ll} \frac{\partial F_n}{\partial h^{j+1,p}_{n}} & = \frac{\Delta z_n} {\Delta t^j} C_n^{j+1} + \Delta z_n \frac{\partial S^{j+1}_{m,n}} {\partial h_{n}^{j+1}} + \frac{K^{j+\kappa}_{{n}-1/2}} {0.52\left(\Delta z_{{n}-1} + \Delta z_{n}\right)} + \frac{K^{j+\kappa}_{{n}+1/2}} {0.5\Delta z_{n} + cK^{j+\kappa}_{n+1/2}}\\ & -\kappa \frac{\partial K^{j+\kappa}_{{n}-1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_{n-1} - h^{j+1}_{n}}{0.5\left(\Delta z_{n-1} + \Delta z_{n}\right)}+1\right) + \kappa \frac{\partial K^{j+\kappa}_{n+1/2}} {\partial h^{j+1}_{n}} 0.5\Delta z_n \left(\frac{h^{j+1}_{n}-\left(\phi - z_n\right)}{\left(0.5\Delta z_n + cK^{j+\kappa}_{n+1/2}\right)^2}\right)\\ \end{array} \tag{D.7}\]

Seepage face

\[ \begin{array}{ll} h^{j+1}_n + 0.5\Delta z_n < 0 \to \frac{\partial F_1}{\partial h^{j+1}_{1}} & = \frac{\Delta z_1} {\Delta t^j} C_1^{j+1} + \Delta z_1 \frac{\partial S^{j+1}_{m,1}} {\partial h_{1}^{j+1}}\\ & +\frac{K^{j+\kappa}_{1+0.5}} {0.5\left(\Delta z_{i}+\Delta z_{i+1}\right)} + \kappa \frac{\partial K^{j+\kappa}_{1+0.5}} {\partial h^{j+1}_{1}} \left(\frac{h^{j+1}_1 - h^{j+1}_2}{0.5\left(\Delta z_1+\Delta z_2\right)}+1\right)\\ h^{j+1}_n + 0.5\Delta z_n \ge 0 \to \frac{\partial F_n}{\partial h^{j+1,p}_{n}} & = \frac{\Delta z_n} {\Delta t^j} C_n^{j+1} + \Delta z_n \frac{\partial S^{j+1}_{m,n}} {\partial h_{n}^{j+1}} + \frac{K^{j+\kappa}_{n-1/2}} {0.5\left(\Delta z_{n-1} +\Delta z_{n}\right)} + \frac{K^{j+\kappa}_{n+1/2}}{0.5\Delta z_n}\\ &- \kappa \frac{\partial K^{j+\kappa}_{n-1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_{n-1} - h^{j+1}_n}{0.5\left(\Delta z_{n-1}+\Delta z_n\right)}+1\right) +\kappa \frac{\partial K^{j+\kappa}_{n+1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_n}{0.5\Delta z_n}+1\right) \end{array} \tag{D.8}\]

Free drainage

\[ \begin{array}{ll} \frac{\partial F_n}{\partial h^{j+1,p}_{n}} & = \frac{\Delta z_n} {\Delta t^j} C_n^{j+1} + \Delta z_n \frac{\partial S^{j+1}_{m,n}} {\partial h_{n}^{j+1}} + \frac{K^{j+\kappa}_{n-1/2}} {0.5\left(\Delta z_{n-1} +\Delta z_{n}\right)} + \frac{K^{j+\kappa}_{n+1/2}}{0.5\Delta z_n}\\ &- \kappa \frac{\partial K^{j+\kappa}_{n-1/2}} {\partial h^{j+1}_{n}} \left(\frac{h^{j+1}_{n-1} - h^{j+1}_n}{0.5\left(\Delta z_{n-1}+\Delta z_n\right)}+1\right) +\kappa \frac{\partial K^{j+\kappa}_{n+1/2}} {\partial h^{j+1}_{n}} \end{array} \tag{D.9}\]

The internodal conductivity \(K^{j+1}_{n+1/2}\) is in the case of free drainage irrespective the value of always treated implicitly.